Sri Lankan Journal of Physics, Vol. 1 (2000) 43-55
Institute of Physics - Sri Lanka
Scattering of a dyon from hydrogen atom
P.C. Pant, V.P. Pandey and B.S. Rajput*
Department of Physics, Kumaun University, Nainital-263 002, India
Abstract
Undertaking the scattering of a dyon from a hydrogen atom it has been demonstrated that
scattering cross section is perceptibly modified from the scattering cross section of
scattering of an electron from a hydrogen atom due to the presence of magnetic charge on
dyon and high energy of dyon involved in the scattering process.
1. INTRODUCTION
Physicists have long been interested in the existence of magnetic monopole. The
early historical interest in monopoles was due to the symmetry between electric and
magnetic fields in Maxwell's equation. However, due to lack of abundance of free
magnetic charge compared to electric charges they were not included in the final
formulation of those equations. In 1931 Dirac1 showed that existence of free magnetic
charge (Dirac monopole) could provide reason for quantization of electric charge2. This
work motivated renewed interest in searching for monopoles. Although there was no
guidance as to the mass, size, etc. of these monopoles several experimental consequences
were apparent.
It was assumed that the monopole mass would not be very much different from
other elementary particles (e. g. protons) and would be highly relativistic. As such these
*
Corresponding author
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
would produce a great deal of ionization while passing through matter but none of these
effects were observed and literature turned partially negative casting doubt on the
existence of these particles.
A fresh interest in the subject was enhanced when 't. Hooft3 and Polyakov4
demonstrated separately that monopoles exist as solutions in many non-Abelian gauge
theories. The possibility of these GUT monopoles provide stimulus for much recent
interest in the subject. These monopoles have enormous importance in connection with
the problem of quark confinement 5 of quantum chromodynamics, C.P. violation6, Proton
decay7 and baryon number non-conservation processes8. Inspite of potential importance
of these particles the theories to describe them suffered from many paradoxes such as
Dirac's veto and wrong connection between spin and statistics9. Schwinger10 showed that
some of these problems can be resolved by taking electric and magnetic charge on the
same particle known as dyon. Moreover, Witten has shown that monopoles are
necessarily dyons6 .
The theories to describe these particles were also clumsy and manifestly noncovariant. In order to develop a theory for these particles which will be conceptually as
transparent as the usual quantum electrodynamics is, we 11,12,13started with the idea of
two four-potentials to avoid the use of singular potential by taking generalized charge,
generalized four potential, generalized four-current associated with these particles as
complex quantities with their real and imaginary parts as electric and magnetic
constituents. With the help of this theory we have undertaken the study of bound states
and scattering of dyon-dyon14 and dyon-fermion 15 systems and it has been demonstrated
that exact solutions of bound states for these system in relativistic frame work is not
possible due to the presence of a term vanishing more rapidly than r-1 in the potential of
such system. To overcome this difficulty we studied the Pauli equation for dyon-dyon16
and dyon-fermion 17 system by adhoc introduction of spin in the Hamiltonian of the
system and obtained bound state solutions in Abelian and non-Abelian gauge theories.
We have further studied the bound states of three and four dyons 18,19 and have
demonstrated that the bound state solutions are quite modified from the bound state
solutions of quantum electrodynamics due to the presence of magnetic charge on dyon.
Extending this work in the present paper we have undertaken the problem of scattering
of an energetic dyon from a Hydrogen atom and it has been demonstrated that scattering
cross section is perceptibly modified from the usual scattering cross section of a fermion
from Hydrogen atom due to the presence of magnetic charge on dyon.
44
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
2. SCATTERING OF A DYON FROM A HYDROGEN ATOM
In order to undertake the scattering of a dyon from a Hydrogen atom we assume
that atom has infinite degrees of freedom, so that it can be excited during the scattering
process. The incident dyon may change place with the electron of atom and hence
exchange effects may occur in the collision. The incident dyon produces generalized
electromagnetic field which may polarize the target atom and hence polarization effects
are also involved .
If we consider the energy of incident dyon as very high the exchange and the
polarization effects are unimportant and can be left out of consideration.
The Hamiltonian for describing the scattering of a dyon by a Hydrogen atom may
be written as
$ = H
$ + H
$′
H
...(2.1)
o
2
2
2
$2 − h ∇
$2 − e
$ = − h ∇
where H
o
r1
2m 1
2m 2
describes the internal motion of the atom together with the kinetic energy of the relative
motion of the incident dyon and the scatterer atom and
2
2
2
2
$ ′ = − e + (eg) − e + (eg)
H
2
r2
r12
2 mr12
2 mr22
...(2.2)
represents the interaction between the incident particle and the scatterer. Hence equation
(2.1) can be written as
eg) 2
eg) 2
(
(
h2 $ 2
h2 $ 2
e2
e2
e2
$
∇ −
∇ −
−
+
−
+
H = −
2
2m 1
2m 2
r2
r1
r12
2 mr22
2 mr12
...(2.3)
$ are specified by two parameter
The eigenfunctions of H
o
r r
r r
$ φ
H
o αa r1 , r2 = E αa φ αa r1 , r2
(
)
(
)
...(2.4)
Here α - specifies the initial quantum state of the incident dyon and a-specifies that of
the atom. We can write
r r
r
r
φ αa r1 , r2 = w a ( r1 )φ α r2
...(2.5)
(
)
( )
45
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
r
r
where w a ( r1 ) is the unperturbed wave function for the atom and φ α ( r2 ) = exp
r r
ik α⋅ r2 is the free particle wave function for the incident dyon.
(
)
E αa = E α + ∈a
...(2.6)
where, E α is the kinetic energy of the incident dyon and ∈a is the unperturbed
eigenvalues of the atom.
( +)
$ in the Born
We can write the wave function ψ αa of the total Hamiltonian H
approximation as follows
2m
r r
( +) r r
(+) r r r r
ψ αa r1 , r2 = φ αa r1 , r2 +
G αa r1 , r2; r1′ , r2′
∫
h2
(
)
(
(
)
(
)
) (
)
r r
(+) r r
H ′ r1′ , r2 ′ ψ αa r1′ , r2 ′ d 3 r1′d 3 r2′
...(2.7)
( +) r r r r
where G αa r1 , r2; r1′ , r2′ is the Green's function for the solution (2.6).
(
)
If the velocity of incident dyon is very high, we can use the Born Approximation
( +)
(replacing ψ αa inside in eqn. (2.7) by the free particle function φ αa, ) and hence, we
( +)
can write the asymptotic behaviour of ψ αa as follows
exp ik α r2 r
r
r r
r
( +) r r
f k β , b; k α , a w a r1
ψ αa r1 , r2 ⎯ ⎯⎯⎯⎯⎯→ φ αa r1 , r2 + ∑
r2 → ∞
r2
b
(
)
(
(
)
)
(
)
( )
...(2.8)
where
(
)
r
r
1
2m
⋅
f k β , b; k α , a = −
4π h 2
∫∫
(
)
r
r
r
r r
exp ik β ⋅ r2′ w ∗b r1′ H ′ r1′ , r2′
(
( ) (
)
)
r r
φ αa r1′ , r2′ d 3 r1′d 3 r2′
= −
2m
4 πh 2
∫∫
(
)
r
r
r
r r
exp − ik β ⋅ r2′ w ∗b r1′ H ′ r1′ , r2′
r
r
exp ik α ⋅ r2′
(
)
( ) (
)
( )
r
w a r1′ d 3 r1′d 3 r2′
...(2.9)
46
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
or
(
)
r
r
2m
f k β , b; k α , a = −
4 πh 2
r
∫∫ exp(ik ⋅ r2 )w b ( r1 )H ′( r1′, r2′ )
∗ r
r
r r
( )
r
w a r1′ d 3 r1′d 3 r2′
where
r
r
r
K = kα − kβ
(
...(2.10)
)
or, deleting primes on the variables of integrations, this can also be written as:
(
)
r
r
2m
f k β , b; k α , a = −
4 πh 2
∫∫
r r
r
r r
exp ik ⋅ r2 w ∗b r1 H ′ r1 , r2
(
) ( ) (
)
( )
r
w a r1 d 3 r1d 3 r2
...(2.11)
Now, for considering dyon scattering from a Hydrogen atom we start with the following
Hamiltonian
2
2
(eg) 2
(eg) 2
r r
e
e
H ′ r1 , r2 = −
+
−
+
...(2.12)
2
r12
r2
2 mr12
2 Mr22
(
)
Thus the scattering amplitude (2.11) can be written as
2m
f = −
4 πh 2
∫∫
2
r r ⎡ 2 ⎧⎪ 1
1 ⎫⎪ (eg)
⎢
+
exp ik ⋅ r2 − e ⎨
⎬+
⎢
r2 ⎪⎭
2M
⎪⎩ r12
⎣
(
)
( )
⎧
⎫⎤
1 ⎪⎥
⎪ 1
⎨ 2 + 2⎬
r2 ⎪⎭⎥⎦
⎪⎩ r12
( )
r
r
w ∗b r1 w a r1 d 3 r1d 3 r2
...(2.13)
Solving this equation in the usual way, we get
f =
2 me 2
h2k 2
2
m(eg)
+
Mh 2 k
r
r 3
∗ r
∫ [1 − exp(ik ⋅ r1 )]w b ( r1 )w a ( r1 )d r1
r
∫[
)] ( )
r r
r
r
1 − exp ik ⋅ r1 w ∗b r1 w a ( r1 )d 3 r1
(
...(2.14)
For
r elastic
r scattering, the initial and final state of the atom are the same i.e. wa ≡ wb and
k α = k β follows from the energy conservation. Therefore, we may write scattering
amplitude as
47
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
f el =
+
2 me 2
h2 k 2
∫ [1 − exp(ik ⋅ r1 )]w a ( r1 )w a ( r1 )d
r
r
∗ r
r
3r
1
2
r r
m(eg)
r
r
1 − exp ik ⋅ r1 w ∗a r1 w a ( r1 )d 3 r1
2
Mh k
[
)] ( )
(
...(2.15)
If we consider ground state of the atom the scattering cross section is given as\
f el (θ) =
−2 ⎤ ⎡
k( eg) 2
2m ⎡
1
⎢1 − ⎛⎜1 + a o2 k 2 ⎞⎟ ⎥ ⎢e 2 +
⎝
⎠ ⎥⎢
4
2M
h 2 k 2 ⎢⎣
⎦⎣
⎤
⎥
⎥
⎦
...(2.16)
In the high energy case where Born approximation is valid the scattering amplitude is
2m
f el ⎯ ⎯⎯⎯⎯⎯→
K l arg e
h2 k 2
⎡
k(eg) 2
⎢e 2 +
2M
⎢
⎣
⎤
⎥
⎥
⎦
...(2.17)
For inelastic case, wa is different from wb and hence the first term of both the integrals
in equation (2.14) are zero because of orthogonality of unperturbed states of atom. So,
we have
r r ∗ r
2 me 2
r
f inel. = −
exp
ik
⋅ r1 w b r1 w a ( r1 )d 3 r1
∫
2
2
h k
(
m(eg) 2
−
Mh 2 k 2
∫
) ( )
r r
r
r
exp ik ⋅ r1 w ∗b r1 w a ( r1 )d 3 r1
(
) ( )
...(2.18)
Let us consider the case when the ground state of atom (1s - state) is excited to the 2s
state (first excitation) due to collision with incident dyon. The scattering amplitude for
this scattering is given by
f inel (1s → 2s) = −
k(eg) 2
2 m ⎡⎢ 2
e +
2M
h 2 k 2 ⎢⎣
⎤
r
⎥ exp ik ⋅ rr1 w ∗ rr w ( rr1 )d 3 r1
∫
2s 1 1s
⎥
⎦
(
) ( )
...(2.19)
48
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
For Hydrogen atom we have
1
⎫
⎪
⎪⎪
⎬
1
⎞
⎛
⎞
⎛
r
r
⎟⎪
⎟ exp⎜ −
⎜2 −
w 2s =
a
a
⎠⎪
⎝
⎠
⎝
0
0
4 2 πa 30
⎪⎭
exp⎛⎜ − r a ⎞⎟
⎝
3
0⎠
πa 0
w1s =
where a o =
...(2.20)
h2
Me 2
There fore,
2m
f inel (1s → 2s) = −
h2 k 2
∫
2
⎡⎧
⎢⎪⎨e 2 + k(eg)
⎢⎪
2M
⎣⎩
⎤
⎫
1
⎪
⎥
⎬
3
⎪⎭ 4 π 2 a o ⎥⎦
r r
⎛
⎛ 3 r ⎞ 3
r ⎞
⎟⎟ exp⎜⎜ −
⎟⎟ d r
exp ik ⋅ r exp⎜⎜ 2 −
ao ⎠
⎝
⎝ 2 ao ⎠
(
)
...(2.21)
The above equation gives us
−8 2 ⋅ a o2
⎡
k (eg) 2
2
⎢e +
f inel =
3 ⎢
2M
9
⎛
⎞
h 2 ⎜ + k 2 a o2 ⎟ ⎣
⎝4
⎠
⎤
⎥
⎥
⎦
...(2.22)
3. EXCHANGE SCATTERING OF DYON FROM HYDROGEN ATOM
In exchange scattering, the dyon takes the place of electron of the atom (having
coordinate r1) and the electron of the atom is scattered. In this case, coordinate of the
incident dyon r2 and coordinate of the electron of the atom (as shown in fig. 1) i.e. r2 are
interchanged after scattering has taken place and hence the final state φ βb for the
exchange scattering can be written as
r
r r
φ exch. r1 , r2 = exp ik β ⋅ r1 w b r2
...(3.1)
βb
(
)
[
] ( )
49
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
( +) r r
Also, the asymptotic behaviour of ψ αa r1 , r2 is given by
(
( +)
ψ
)
(
)
(
)
exp ik β r1 r
r
r r
r
r
,
r
⎯
⎯⎯⎯⎯
⎯
→
g
k
,
b
;
k
, a w b r2
∑
αa 1 2
β
α
r1 → ∞
r1
b
(
)
( )
...(3.2)
There is no scattered
r
r part here because the dyon is captured in the atom. The scattering
amplitude g k β , b; k α , a in the exchange scattering is given by
r
r
2m
r r
r r
( +) r r
g k β , b; k α , a = −
φ exch.* r1 , r2 H ′ r1 , r2 ψ αa r1 , r2 d 3 r1d 3 r2
∫∫
b
β
4 πh 2
(
(
)
)
= −
(
2m
4 πh 2
∫∫
) (
)
(
)
r
r r
( +) r r
exp ik β ⋅ r1 w ∗a r2 H ′ r1 , r2 ψ αa r1 , r2 d 3 r1d 3 r2
(
) ( ) (
)
(
)
...(3.3)
Since, we are assuming that the incident dyon is a fermion so that total wavefunction
must be antisymmetric. In non-relativistic limit the wavefunction can be written as
follows
ψ(
+) r r r r
+ r r
+ r r
r1 , r2 ; s1 , s2 = ψ ( ) r1 , r2 χ ( ) s1 , s2
...(3.4)
+ r r
+ r r
where ψ ( ) r1 , r2 is the space part and χ ( ) s1 , s2 is spin part of the total wave
+ r r
function. Since χ ( ) s1 , s2 describes two spin half particles, we can have either a
singlet state or a triplet state. The singlet state is
1
( +) r r
χ sin g. s1 , s2 =
...(3.5)
{α(1)β(2) − α(2)β(1)}
2
(
(
)
)
(
(
(
)
(
)
(
)
)
)
which is antisymmetric. Thus to make the total wave function antisymmetric, we symmetrize the space
part
( +) r r
(+) r r
+) r r
ψ (sym
. r1 , r2 = ψ αa r1 , r2 + ψ αa r2 , r1
(
)
(
)
(
)
...(3.6)
The first function on the right hand side corresponds to the direct scattering and the
second one corresponds to exchange scattering. Asymptotic behavior of the symmetrized
wavefunction (3.6) can be written as
+
ψ( )
(
)
exp ik β r1
r r
r r
r
⎯→ φ αa r1 , r2 + ∑
( f + g)w b r2
sym. r1 , r2 ⎯ ⎯⎯⎯⎯
r1 → ∞
r1
b
(
)
(
)
50
( )
..(3.7)
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
Thus the scattering amplitude in the singlet state is the sum of the direct and exchange
scattering amplitudes and hence the scattering cross section for singlet state is given by
kβ
σ sin g. =
kα
f +g
2
...(3.8)
The triplet state is given by
⎧
α(1)α( 2)
⎪
⎪⎪
( +) r r
χ trip. s1 , s2 = ⎨
β(1)β( 2)
⎪ 1
⎪
{α(1)β(2) + α(2)β(1)}
⎪⎩ 2
(
)
...(3.9)
i.e. all the triplet states are symmetric. Thus to make the total wave function antisymmetric, we make the space part antisymmetric
r r
( +)
( +) r r
( +) r r
ψ antisym. r1 , r2 = ψ αa r1 , r2 − ψ αa r2 , r1
...(3.10)
(
)
(
)
(
)
In this case, scattering amplitude will be f − g and hence the scattering cross-section
in triplet state is given by
kβ
2
σ trip. =
...(3.11)
f −g
k
α
Equation (3.8) and (3.11) give the differential scattering cross section including
exchange effect. Total differential scattering cross section is the sum of the σ sin g. and
σ trip. with their proper statistical weight factor
1
3
⎛ dσ ⎞
=
σ sin g. + σ trip.
...(3.12)
⎜
⎟
⎝ dΩ ⎠ tot.
4
4
In order to evaluate equation (3.12) we should calculate f and g. The direct scattering
amplitude f has already been calculated and is given by equation (2.14) while g is given
by
r
2m
r
r
r r
( +) r r
g = −
−
ik
⋅ r1 w ∗b r2 H ′ r1 , r2 ψ αa r1 , r2 d 3 r1d 3 r2
exp
β
∫∫
4 πh 2
(
) ( ) (
)
(
)
...(3.13)
To calculate g, we make use of Born-Oppenheimer-approximation. In this
r r
( +) r r
approximation, we replace ψ αa r1 , r2 by the state φ αa r1 , r2 ; the unperturbed
wave function before scattering
(
(
)
51
)
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
g = −
1
2m
⋅
4π h 2
∫∫ exp(−ik β
r
)
r
r
r
r r
r
⋅ r1 w ∗b r2 H ′ r1 , r2 exp ik α ⋅ r2 w a r1 d 3 r1d 3 r2
( ) (
) (
) ( )
...(3.14)
The initial and the final states of the atom are not orthogonal to each other. Due to this, a
number of defects are incorporated into Born-Oppenheimer approximation. To overcome
these difficulties we use another approximation which is due to Ochkur. In this
approximation, g is expanded in the inverse powers of k α and the leading term is only
retained, We have
g ≡ g ee + g ne
...(3.15)
Where g ee is the contribution due to dyon-electron interaction and g ne is the
contribution due to nuclear interaction.
We have
r
r
2 me 2
1
r
r
r
r
g ee = −
exp − ik β ⋅ r1 w ∗b r2
exp ik α ⋅ r2 w a r1 d 3 r1d 3 r2
∫∫
r12
4 πh 2
+
m(eg)
2
2 πMh 2
∫∫
{
}
( )
{
}
( )
(
r
r
r
exp − ik β ⋅ r1 w ∗b r2
) ( )
r
1
r
r
exp ik α ⋅ r2 w a r1 d 3 r1d 3 r2
2
r12
(
) ( )
...(3.16)
Now
1
r12
=
1
2π 2
( (
r r
r
exp iS r2 − r1
S2
∫
)) d 3S
and hence
g Ochkur = −
(
)
+
2 me 2
⋅
h2
m(eg)
1
2
Kα
2
πMh 2
⋅
∫
1
Kα
r
r
r
r
exp ik α ⋅ r2 w ∗b r2 w a r2 d 3 r2
(
∫
) ( ) ( )
r
r
r
r
exp ik α ⋅ r2 w ∗b r2 w a ( r2 )d 3 r2
(
) ( )
...(3.17
with the help of above equation and from the direct scattering amplitude (2.14). we get
2
⎤
m( eg)
Kα
2 me 2
k2 ⎡
⎢
g Ochkur =
(
) k 2 ⎢f − h 2 k 2 δ ba + πMh 2 ⋅ K δ ba ⎥⎥
α ⎣
⎦
52
...(3.18)
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
For inelastic scattering a ≠ b . Therefore δab = o and hence we get
g Ochkur =
(
)
k2
⋅f
2
Kα
...(3.19)
For elastic scattering, a = b i.e. δ ba = 1 . Therefore
k2
g (Ochkur ) =
2
kα
⎡
2
m(eg) 2 K α ⎤
⎢ f − 2 me +
⎥
⋅
2
2
2
K
⎢
⎥
h k
πMh
⎣
⎦
...(3.20)
Now, for elastic scattering of dyon with hydrogen atom.
k( eg) 2
2m ⎡
1 2 2 ⎞ −2 ⎤ ⎡⎢ 2
⎛
⎢1 − ⎜1 + a o k ⎟ ⎥ e +
f =
⎝
⎠ ⎥⎢
4
2 Mπ
h 2 k 2 ⎢⎣
⎦⎣
⎤
⎥
⎥
⎦
For Born approximation
2m
f =
h2k 2
⎡
k(eg) 2
⎢e 2 +
πM
⎢
⎣
⎤
⎥
⎥
⎦
...(3.21)
Therefore
2
Kα
1 ⎡⎢ 2 m(eg) ⎛
g (Ochkur ) =
⎜1 +
2 ⎢
K
k α ⎣ πMh 2 k 2 ⎝
⎤
⎞⎥
⎟
⎠⎥
⎦
...(3.22)
Hence the elastic differential scattering cross section for dyon-atom scattering in the
exchange effect is given by
3
1
2
2
⎛ dσ ⎞
f + g Ochkur +
f − g Ochkur
=
⎜
⎟
⎝ dΩ ⎠ tot.
4
4
=
m(eg) 2
me 2
−
h2 k 2
πMh 2
⎡ K − k2
⎢ α
⎢⎣ KK α
53
⎤
⎥
⎥⎦
...(3.23)
P.C. Pant et al./Sri Lankan Journal of Physics, Vol.1 (2000) 43-55
Fig. 1 Index ‘A’ refers to incident d yon and indices ‘1’ and ‘2’ refers to proton and electron of the
Hydrogen atom respectively.
4. DISCUSSION
Hamiltonian (3.1) describes the dynamics for scattering of a dyon by a Hydrogen
atom. Equation (3.7) is the wave function for the total Hamiltonian under Born
approximation and scattering amplitude for this case is given by equation (3.10).
Equation (3.16) describes the scattering amplitude under Born approximation for elastic
scattering . Scattering amplitude for elastic scattering given by equation (3.17). Equation
(4.8) and (4.11) give the differential scattering cross section including exchange effect
i.e. where electron of hydrogen atom is replaced by the incident dyon in the scattering
process. Total differential scattering cross section is given by equation (4.12). Equation
(4.22) describes the total scattering cross section for scattering of a dyon by hydrogen
atom with inclusion of exchange effect. All of these scattering cross sections are
modified from the usual scattering cross section of scattering of a fermion on the atom
due to the presence of magnetic charge on dyon. These scattering cross sections reduce
to the usual scattering cross sections of scattering of a fermion from an atom in the
absence of magnetic charge.
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